Skills, Agglomeration and Segmentation

Tomoya Mori∗and Alessandro Turrini

June 2001

Abstract

We investigate the role of skill heterogeneity in explaining location patterns induced by pecuniaryexternalities (Krugman [30]). In our setting, sellers with higher skills better perform in the mar-ketplace, and their sales are larger. Selling to distant locations leads to lower sales because of both(pecuniary) transport costs and communication costs that reduce the perceived quality of goods. Asymmetry-breaking result is obtained: symmetric configurations cannot be stable, and regional in-equality is inevitable. The relatively more skilled choose to stay in the location with higher aggregateincome and skill, while the relatively less skilled in the other. The model allows us to analyze thelinks between the extent of interregional inequality and the extent of interpersonal skill inequality.

Keywords: Skill heterogeneity; Agglomeration; Core-periphery model; Regional inequality; Interpersonalinequality; Transport and communication costs.

JEL ClassiÞcation: R12, R13, F12, F16.

Acknowledgments: We thank Richard Arnott, Duncan Black, Pierre-Philippe Combes, Gilles Du-ranton, Masa Fujita, Tatsuo Hatta, Gianmarco Ottaviano, Diego Puga, Akihisa Shibata, Tony Smith,Takaaki Takahashi and participants of the seminars/conferences for their constructive comments and en-couragement. A part of the research was conducted while we were visiting CORE, Universite catholiquede Louvain, Belgium, where we beneÞted from the stimulating discussions with Jacques Thisse. Thisresearch has been supported by the Fonds National de la Recherche ScientiÞque of Belgium, the NomuraScience Foundation, the Grant in Aid for Research (Nos. 09CE2002 and 09730009) of the Ministry ofEducation, Science and Culture in Japan.

∗Institute of Economic Research, Kyoto University, Yoshida-honmachi, Sakyo, Kyoto, 606-8501 Japan. Phone: +81-75-753-7121; Fax: +81-75-753-7198; E-mail: mori@kier.kyoto-u.ac.jp.

Department of Economics, University of Bergamo, Piazza Rosate 2, 24100 Bergamo, Italy. Phone: +39-35-277506;Fax: +39-35-249975; E-mail: alessandro.turrini@uni-bocconi.it.

1 Introduction

Economic activities are not evenly distributed in space. Some places are crowded with Þrms and workers,

while others, poor of people and human capital, lag behind. The economic landscape is full of humps

and bumps, that result from an unequal distribution of labor across space, both in terms of its quantity

and quality. We generally observe the most educated and talented workers clustering together in wealthy

and economically important areas, while low skilled workers are more often located in low income areas.

Which factors account for the observed agglomeration of workers, and for their spatial segmentation across

skill and income levels? This paper studies the role of skill heterogeneity in new economic geography

models of location.

The spatial distribution of human capital is far from being uniform. Workers tend to cluster together,

the higher their level of human capital and skill. Figure 1 helps to illustrate the point. There, we report

the Lorenz curves of the inter-city distribution of workers in Japan, year 1990.1 Lorenz curves are depicted

separately for the total population and for the workers belonging to particular education categories.2 The

Þgure clearly indicates that workers with higher education levels are more geographically concentrated.3

The data also show that workers with higher education are more easily found in big cities: the rank

correlation between city size and the share of college graduates in a city is 0.73, while that between city

size and the share of workers with at most a junior high school degree is -0.67. Tokyo alone accounts

for more than 40% of college graduates among all metropolitan areas. These patterns are not typical of

Japan: similar evidence is obtained for the case of the US.4

Figure 1

In this paper we present a model for the location decisions of workers endowed with different skill

levels that produces results consistent with available evidence.

The lumpy interregional distribution of human capital results from the location decisions of workers,

which may differ according to their skill levels (based on their education, experience and innate ability).

The existing distribution of skill affects in turn the outcome of the migration choices of workers.5 Two

1Data source: Japan Statistics Bureau [29, Vol.3, Part 2]. A city here is a Standard Metropolitan Employment Area(SMEA) which is an aggregation of counties based on the commuting pattern. For the precise deÞnition of SMEA, seeYamada and Tokuoka [42].

2The horizontal axis ranks cities in terms of their shares of workers with a given education level (the rank is normalizedby the total number of cities, 124). The vertical axis reports the cumulated share of workers of a given education level inthe cities up to a given position in the rank.

3The value of the Gini index for the distribution of the total population is 0.69, while the corresponding value for thecases of workers with education up to junior high school, high school, community/technical college, and college, is 0.60,0.69, 0.77, and 0.82 respectively.

4In 1990, the Gini index value for the inter-city distribution of workers is 0.61, while it is 0.66 for the case of collegegraduates (aged above 25). The rank correlation between city size and college graduates share in the city workforce is 0.40.These Þgures are calculated from Black [5]s data set which is based on the Population Census of the US.

5Borjas, Bronars, and Trejo [8] and Rauch [37] give empirical support to the view that these patterns may be due to the

1

approaches to study location decisions and the interregional distribution of workers have received the

greatest favor. The Þrst puts at the centre of agglomeration forces the existence of knowledge spillovers,

the second deals with the existence of pecuniary externalities arising from the interaction of increasing

returns to scale with transport costs. The analysis of economic agglomeration as an outcome of knowledge

spillovers goes back to Marshall [33].6 Recent theoretical explanations for interregional sorting of workers

according to their skill levels are found in Black [5] and Black and Henderson [7].7

The interaction between scale economies and transport costs has a long-standing tradition in spa-

tial economics, but the study of agglomerations on the basis of pecuniary externalities became part

of mainstream analysis only after the core-periphery model by Krugman [30].8 This model offers a

comprehensive and analytically convenient representation of how the interaction among scale economies,

transport costs and an immobile source of demand generates centripetal and centrifugal forces. However,

the existing versions of the core-periphery model do not account for the location patterns of economic

agents of different skill, leaving unexplained the observed tendency towards spatial segmentation accord-

ing to skill levels.

As for empirical evidence, though there is a consensus about the pervasiveness of positive externalities

that generate agglomeration forces in reality, it is quite difficult to disentangle empirically between the

different sources of externalities that explain the observed agglomeration patterns.9

In this paper, we analyze the spatial distribution of human capital on the ground of pecuniary exter-

nalities without resorting to knowledge spillovers, offering an alternative explanation for the interregional

sorting of workers according to skill levels.10 We reinterpret the familiar core-periphery model, consid-

ering mobile worker-sellers whose earnings consist of rents associated with their skills. The economy

we have in mind is made up of many, heterogenous sellers who offer goods that are differentiated both

horizontally (variety) and vertically (quality), where quality requires skills. Selling to distant markets

entails a physical transport cost: a fraction of the shipped good melts away. This is a basic feature

of the core-periphery model, and a common one for almost all new economic geography models. In our

model, however, selling to faraway consumers leads to an additional communication cost, that shows up

workers tendency to self-select according to their skill levels across locations. The difference between ones own skill leveland the average skill level of ones initial home town appears to be indeed an important explanatory factor for individualmigration decisions.

6See also Jacobs [28] and Lucas [31].7See also Benabou [3][4] for models of intra-urban spatial segregation based on human capital externalities.8Surveys can be found in Fujita [20] and Fujita, Krugman and Venables [21].9Davis and Weinstein[13] Þnd evidence of a signiÞcant home market effect, namely, a more than proportional relation

between local production and local demand implied by the existence of scale economies and factor mobility. Rauch [37]presents results in favor of the existence of knowledge spillovers, showing that the US workers are more productive in citieswhere average levels of educational attainment are higher. Results similar to Rauch [37] are found in Glaeser, Sheinkman,and Shleifer [26], Ciccone and Hall [12], and Dobkins and Ioannides [16], while Ciccone[9], Ciccone and Peri[10] and Ciccone,Peri and Almond [11] reject the hypothesis of signiÞcant human capital externalities. Finally, Dumais, Ellison and Glaeser[17] Þnd empirical conÞrmation of both pecuniary externalities and knowledge spillovers, but claim that labor marketpooling have the major role in explaining agglomeration patterns.

10See also Fernandez[18] for a review of models that explain within-cities spatial sorting according to skill levels on theground of heterogeneity in preferences for local public goods.

2

as a reduction in goods quality. One can think of many instances. Buyers may know the quality and

characteristics of goods more precisely through direct contact rather than through mailing or advertising.

The use of complex goods may require strict interaction between buyers and sellers, and interaction is

hampered by distance. In general, we observe empirically a strong home bias in consumption: trans-

port and trade costs are not enough to explain the preference of consumers for homemade goods (see,

e.g., Helliwell [27]).11

Why does the presence of communication costs matter in our version of the core-periphery model?

The reason is that they lead to a different impact of distance on the behavior of workers depending

on their skill level. Because sellers of higher skill perform better in the marketplace and obtain larger

demand for their goods, they will normally be hit relatively less by distance from the market. Real-life

instances seem supportive of this view. In general, we see that only the most successful professionals and

entrepreneurs start selling their goods and services in faraway markets. When only transport costs of the

iceberg type are present, however, selling to a distant location corresponds to a constant share of sales

(and proÞts) lost in transit. This means that more skilled and less skilled sellers would be affected in the

same proportion by distance.12 Adding communication costs to the core-periphery structure turns out

to be a very convenient way to avoid costs of distance that are perfectly proportional to the value of the

shipped goods, as it is the case when only transport costs of the iceberg type are present.13 This way,

the more skilled are somehow closer to the global market, and, as in Rosens [38] Superstar economy,

that an increase in the economy-wide demand raises disproportionately the rents associated with a higher

level of skills. In this context, since the earnings of agents with higher skills are less dependent upon

local markets, they are relatively more footloose than the less skilled. This is the feature that explains

self-sorting according to skill levels in our model. The more skilled are found in locations where local

competition is tougher, but where the goods are of higher quality and a wider range of varieties is

available, while the less skilled are in locations where competition is low, but where goods are of lower

quality and less variety is offered. Transport costs and regional characteristics (local market size and

cost of living) have a different impact on the location decisions of workers depending on their skill level.

The spatial distribution of population and skill, in turn, affects the market size and the cost of living in

different locations.

One of the major results from the equilibrium analysis is that, in contrast with the predictions of the

core-periphery model and all of its existing variants, the symmetric equilibrium where all locations are

11This observed home bias in consumption justiÞes the rather ad-hoc Armington [1] assumption in applied tradeanalysis.

12Of course, such a feature of iceberg transport costs is immaterial when all agents are homogenous as in the originalversion of the core-periphery model.

13An alternative approach would be to assume pecuniary transport costs that are, for instance, linear in the quantityof the good shipped. However, adopting such an approach in our analysis while keeping tractability of the model wouldnecessarily require sacriÞcing other parts of the model structure (see, e.g., Ottaviano [36, Ch.3]).

3

identical cannot be a stable one. Starting from a symmetric conÞguration, perturbations to the agents

distribution always lead to a self-reinforcing sorting of mobile workers by their skill level. The reason is

that regional variables in this setting are affected not only by the distribution of agents across locations

(as in the standard core-periphery framework), but also by the spatial distribution of skill. In our model,

however, the concentrated conÞguration (all mobile agents in one location) is not the only candidate

stable equilibrium. In fact, spatial segmentation according to skill levels is the typical stable outcome:

the more skilled choose to stay in the location where aggregate skill and income is higher, while the

less skilled stay in the other. Our model can also serve as a tool to analyze the impact of interpersonal

inequality on the spatial distribution of population and skill. We show that an increase in interpersonal

inequality has different effects on the extent of spatial agglomeration depending on the level of transport

and communication costs and on the mass of human capital in the economy.

The basic message of our analysis is that agents heterogeneity in the presence of pecuniary exter-

nalities may work as a source of regional inequality. The possibility of international inequality has been

already explained on the pure ground of pecuniary externalities.14 Our contribution, we believe, is to

show that amending the standard core-periphery model with heterogenous agents is sufficient to predict

the inevitability of regional inequality. Workers segmentation across skill levels can thus be thought of

as a pervasive tendency that does not necessarily require the presence of human capital externalities.

Market interactions among unequal agents can be sufficient to generate spatial sorting according to skill

levels. This naturally leads to consider distributional issues as a key to understanding agglomeration

patterns, and may help in explaining why regional inequalities appear to be associated with interpersonal

income inequality in the real world.15

The remainder of the paper is organized as follows. The next section presents the structure of the

model. Section 3 introduces the notion of temporary equilibrium and adjustment process. Section 4

deÞnes the equilibrium. Section 5 classiÞes possible equilibrium conÞgurations, and is devoted to the

analysis of their existence and stability. Section 6 studies the global dynamics of the economy. Section 7

discusses some implications of our model. The concluding comments end the paper.

14Matsuyama [34] proposes an argument based on increasing returns arising from the trade in inputs and the internationaldivision of labor.

15It is interesting to compare the US and Japan in this respect, the former being one of the most unequal among theadvanced countries (the richest 5% of the population accounts for more than 56% of the national wealth in 1983), and thelatter a relatively equal country (the richest 5% of the population accounts for 25% in 1984, see Wolff [41] for these Þgures.).In the US, the ratio of the real wage in the richest city and that in the poorest city is above 7.0 (Dobkins and Ioannides[15]), while in Japan, this ratio is almost Þve times smaller (Nakamura and Tabuchi [35]).

4

2 The model

2.1 The economy

The economy consists of two regions, denoted by a and b, which are symmetric except for the mass of

skilled workers that are located in each. Variables in the model bear a subscript r = a, b to indicate

the region. Firms produce differentiated goods under imperfect competition and free-entry. Consumers

(skilled and unskilled workers) like variety, according to the Dixit-Stiglitz [14] formulation. There are

two types of primary production factors: skilled and unskilled labor, where each worker embodies a unit

of corresponding labor. The production requires two inputs: skill (talent) provided by the former

and an intermediate input, produced out of the latter. Skilled workers are perfectly mobile between the

regions, while the unskilled are assumed to be immobile. The markets for the production factors and the

intermediate good are competitive.

Unskilled workers are homogeneous. A unit mass of them is located in each of the two regions. Skilled

workers are heterogeneous, in that they are distinguished by different skill levels. There is a unit mass

of skilled workers in the whole economy. Each skilled worker is characterized by her skill level s; a more

talented worker is associated with a larger value of s. The lowest and the highest skill levels among

all skilled workers are denoted, respectively, by s and s, where 0 < s < s < ∞. Skilled workers aredistributed over the interval, S ≡ [s, s], according to the density function f(s). The aggregate (and

average) skill in the economy is denoted by bs, which equals Rs∈S sf(s)ds. No additional assumptions are

made on the distribution of skills. To express the regional distribution of workers of each skill level, let

fr(s) (r = a, b) represent the density of workers with skill level s in region r; denote by nr the share of

skilled population, and by sr the share of aggregate skill in region r. Then we can write:

nr =

Zs∈S

fr(s)ds, sr =1bs

Zs∈S

sfr(s)ds. (1)

wheresa + sb = 1, na + nb = 1, (2)

fa(s), fb(s) ≥ 0 and fa(s) + fb(s) = f(s) for each s. (3)

The intermediate input is produced through a linear technology using unskilled labor as sole input.

Final production requires the services of skilled labor and a given amount of the intermediate input

proportional to output. While the intermediate inputs provide standardized production services, skilled

workers talent adds value, or quality, to each unit of the Þnal good. Goods of higher quality are

more appreciated by consumers. Consequently, we consider products that are differentiated along the

horizontal dimension (variety) and the vertical one (quality).16

The intermediate inputs are costlessly mobile across regions. Shipping Þnal goods instead incurs

iceberg transport cost: a fraction of the good is lost in transit. We further assume that consuming Þnal

16The same formulation is found in Manasse and Turrini [32].

5

goods outside the region in which they are produced incurs an additional communication cost. Namely,

the perceived quality of products is lower if the goods are shipped to the other region.

2.2 Distribution of skills and feasible spatial configurations

In our economy, the interpersonal distribution of skill and the spatial distribution of skilled workers are

necessarily related. A graphical representation of the economy as in Figure 2 helps the intuition.17 In

the graph, the horizontal distance from O [resp., O0] measures the (skilled) population share in region a

[resp., b]. The vertical distance from O [resp., O0] measures the share of aggregate skill in region a [resp.,

b]. The shaded area in each diagram, denoted by M , represents the feasible domain of na and sa. At

each point on the upper [resp., lower] boundary ofM the spatial conÞguration is such that all workers in

region a are at least [resp., at most] as skilled as those in region b. The construction of these boundaries

is as follows. Take any aggregate skill share sa in region a. At the corresponding point on the upper

boundary, region a has the smallest population share consistent with sa. In other words, at each point

on the upper boundary, there exists a threshold skill level z, such that all workers with skill level higher

than z are in region a; those with skill level lower than z are in region b; and those with skill level z

may locate in either of the two regions. The lower is sa, the higher the threshold skill level z: only a

few of the most skilled workers are needed in region a to attain sa. The lower boundary is obtained in a

symmetrical way. In this case, for a given value of sa, we must Þnd the largest population share of region

a consistent with sa.

Figure 2

The boundaries of area M have a straightforward interpretation: each of them is the Lorenz curve

of the interpersonal distribution of skill in the economy. The size of area M corresponds to the value

of the Gini index for the skill distribution. The larger is M , the greater the extent of inequality in the

interpersonal distribution of skills. The standard core-periphery model is a particular case where M has

size zero, and the feasible allocations are restricted on the 45 line which passes through the symmetric

conÞguration at E (na = sa = 1/2). Since na and sa must necessarily be in areaM , it follows also that the

feasible regional allocations of population and skill are necessarily shaped by the extent of interpersonal

inequality in the economy.

2.3 Technology and preferences

The consumption good can be differentiated along a continuum of varieties i ∈ R. Each variety i isproduced out of intermediate inputs and skill. The size of Þrms is normalized in such a way that one Þrm

17For the ease of exposition, in Figure 2, a continuous distribution of the population over the skill range S is assumed.

6

employs one skilled worker only.18 We further assume that each skilled worker can employ her skill in

the production of at most one variety of the consumption good. The intermediate input requirements for

each variety are proportional to output. Let wr(s) and υr denote, respectively, the return to the skill of

a worker endowed with talent s and the marginal cost (consisting of expenses for intermediate inputs)

of Þnal production in region r. Then, the cost, Cr(Q, s), of producing Q units of any variety in region r

by employing a worker with skill level s is given by

Cr(Q, s) = wr(s) + υrQ. (4)

Firms are atomistic proÞt-maximizers. They produce goods which are imperfect substitutes and set their

price taking as given other Þrms choices (the large group Chamberlinian hypothesis holds). Consumer

utility increases with the extent of variety in consumption. As it is standard in monopolistic competition

models, love for variety plus increasing returns in production insure that no Þrm is willing to supply the

same variant offered by a rival. Since each Þrm requires the skill of one worker, we necessarily have the

total variety size equal to 1. In turn, the condition of free-entry ensures that skilled workers receive all

the operating proÞts realized by Þrms. In analogy with Rosen [38], skilled workers income thus consists

of skill rents associated with Þrms operating proÞts.

As for the production of intermediate inputs, we simply assume that one unit of unskilled labor

produces one unit of the intermediate input. It follows from the assumption of perfect competition in the

market of intermediates that wages for unskilled workers in region r equal υr.

We now turn to the problem of consumers. Recall that in our formulation more talented workers

produce a good of better quality, and better quality is appreciated by consumers. For simplicity, and

without affecting our qualitative results, we assume that the skill level of the worker employed for the

production of the good exactly conveys the quality of the good. So, hereafter we use the terms, skill

level and quality, interchangeably. Consumers derive utility from a combination of the quantity of each

variety i, x(i), and the perceived quality of the good of variety i. Because of love for variety, individuals

will smooth their consumption across all available varieties, both domestic and imported. However, in

the case of shipped varieties, the perceived quality level is lower. While for the case of domestic goods

the perceived quality is equal to the skill level s(i) associated with the good, when the good is imported

the perceived quality equals s(i)− c, where c > 0 represents the communication cost. So, the geographicprovenience of goods has a direct consequence on consumer welfare.19 As will be clear in the following

analysis, the presence of communication costs plays a crucial role. The reason is that it introduces a

non-convexity in the technology for selling goods to distant locations. This non-convexity shows up across

18An alternative interpretation is that each worker is running a Þrm. In the remainder of the paper we refer to our mobileagents as workers or worker-sellers.

19Notice the analogy of this speciÞcation of communication costs with that used in many models aimed at explaininglocation patterns on the ground of agents need for direct contact and face-to-face interaction. See Fujita and Smith [22]for a comprehensive survey.

7

sellers, with more skilled agents selling larger amounts of their goods, being thus proportionally less

affected by distance compared with the less skilled. This, in turn, explains a different location behavior

for workers endowed with different skill levels.

A Cobb-Douglas speciÞcation is chosen to nest the quantity x(i) and the quality s(i) in consumers

utility function, while a standard CES speciÞcation is used to nest different varieties.20 Denoting the

mass of varieties produced within and outside region r by Nr and N−r, respectively, the utility level of

a consumer located in region r is given by:

ur =

"Zi∈Nr

s(i)1−ρx(i)ρdi+Zi∈N−r

(s(i)− c)1−ρx(i)ρdi#1/ρ

. (5)

As usual, ρ ∈ (0, 1), and σ = 1/(1−ρ) > 1 is the elasticity of substitution across different varieties. SinceÞrms are atomistic, σ is also the price elasticity of the demand for each variety.

Unlike the models based on the transport cost of only the iceberg type, in our model, the goods are

not traded if the communication costs are too high (c > s). Since the analysis in this case is unnecessarily

involved, we will focus on the case in which all varieties are traded at equilibrium. Notice that our model

has qualitatively the same structure as that of the original core-periphery model, if all mobile agents

are homogeneous, and communication costs are zero. For our purpose, it suffices to slightly perturb the

structure of the core-periphery model while preserving its tractability. Namely, we maintain the following

assumption:21

Assumption 1 0 < c <s.

Indeed, we show that when interpersonal skill inequality is taken into account in the context of the

core-periphery model, the mere presence of communication costs is sufficient to explain the spatial sorting

of population by skill levels.

3 Temporary equilibrium

In this section, we derive the equilibrium conditions under a given regional distribution of workers. These

are obviously the necessary conditions for a long-run equilibrium in which workers have no incentive to

relocate even if they can. In equilibrium, all workers must be choosing to work at the best available

conditions. To derive such conditions, we Þrst deÞne a temporary equilibrium: a state of the economy in

which the opportunities of workers are restricted within the region where they are temporarily located,

i.e., a temporary equilibrium is realized when, given the regional distribution of workers, (i) consumers

maximize utility; (ii) Þrms maximize proÞts; (iii) proÞts are zero; (iv) all workers of the same skill attain

20This speciÞcation is chosen for analytical convenience only. Adopting a different formulation which still implies apositive relation between quality and associated sales would lead to the same results as ours.

21A detailed analysis under c ≥s is available from the authors upon request.

8

the same utility level in their region; (v) all markets clear. The economy is assumed to attain a temporary

equilibrium instantaneously once the regional distribution of workers is given.

Note Þrst that since intermediates are costlessly transportable, it must be that υa = υb = υ. By

choice of numeraire, we Þx υ = 1. Recall also that the Þnal good, instead, is subject to transport costs

of the iceberg type: for one unit to be shipped to the other region, 1− τ units are lost in transit, and afraction τ arrives to distant consumers, where τ ∈ (0, 1).Keeping this in mind, consider the proÞt maximization problem of a Þrm in region r. Since all Þrms

share the same production technology and the price elasticity of demand is constant and equal to σ, all

Þrms set a common mill price given by22

p = σ/(σ − 1). (6)

Thus, we omit henceforth the variety index. Moreover, we see that if product quality matters for Þrms

performance, it matters only in terms of quantities sold.

Expressions for total demand for a good with quality s in region r differ depending on whether the

good is produced locally or in the alternative region. Denoting, respectively, by Xr(s) and X∗r (s) the

demand in region r for a local good and that for a product provided in the alternative region −r, fromthe consumer utility maximization we obtain

Xr(s) = sp−σIrPσ−1

r , X∗r (s) = (s− c)τσ−1p−σIrPσ−1

r , (7)

where Ir is the aggregate income in region r, given by23

Ir = 1 +

Zs∈S

fr(s)wr(s)ds, (8)

and Pr is the price index in region r, given by

Pr ≡ pbs 11−σ

£sr + (s−r − cn−r/bs)τσ−1

¤ 11−σ . (9)

Given the fact that each unit of differentiated good requires one unit of intermediate good, the total

amount of differentiated goods produced in the economy at equilibrium equals the mass of unskilled

labor, that is, two units. However, the market share of each seller depends on her skill level. The more

skilled sellers are able to sell more as (7) indicates. Note also that transport and communication costs

affect Þrms sales to distant regions differently. We observe in (7) that transport costs affect the sales

of worker-sellers of different skills in the same proportion, while communication costs fall proportionally

more on the less skilled.

22As it is always the case with monopolistic competition, iceberg transport costs, and a CES-Dixit-Stiglitz representationof preferences, Þrms set the same markup over marginal costs in all locations (see, e.g., Fujita, Krugman, and Venables [21,Ch.4]). It is important to bear in mind that when we speak about tougher competition in such a framework we refer toa downward shift in the individual demand curve at given price-cost margins.

23The Þrst [resp., second] term in the RHS of (8) is the aggregate income of immobile [resp., mobile] workers in region r.

9

The equilibrium (operating) proÞt, πr(s), of a quality-s Þrm in region r can be easily obtained as

a function of Xr(s) and X∗−r(s). Recalling that the earnings of each skilled worker coincides with the

Þrmss operating proÞt we obtain

πr(s) = wr(s) =1

σ − 1 [Xr(s) +X∗−r(s)]. (10)

The wage rate is thus proportional to the quantity sold. Since by (7) the sales of the more skilled are

less dependent on their local demand, their wage rate is also less dependent on their location. By solving

(7)-(10) for IrPσ−1r which if multiplied by p−σ is the local demand for a Þrm offering a standard good

(i.e., of quality 1) we obtain

IrPσ−1r =

A−r +BrAaAb −BaBb , (11)

where Ar ≡ P 1−σr − 1

σp1−σsrbs and Br ≡ 1

σ (p/τ)1−σ(srbs − cnr). It can be shown that Ar > Br > 0,

which in turn implies AaAb − BaBb > 0. Thus, two types of local Þrm demand, Xr(s) and X∗r (s), are

determined uniquely and are strictly positive. It follows from (10) that the temporary equilibrium wage

rate wr(s) is also determined uniquely and is strictly positive. The temporary equilibrium utility level of

workers endowed with skill s and located in region r corresponds to their real wage, namely, their wage

deßated by the locally prevailing cost of living index:

ur(s) = wr(s)/Pr. (12)

4 Equilibrium

A temporary equilibrium is not necessarily an equilibrium of the economy in the long-run. Since skilled

workers are freely mobile, they will search for the best opportunities available in the whole economy,

moving across regions if proÞtable. Formally, workers are assumed to migrate toward the region that

gives them a higher utility level. The adjustment process is assumed to be myopic and given by:

úfa(s) = Φ(s|u, n) and úfb(s) = −Φ(s|u,n) (13)

subject to (3), where úfr(s) is the time derivative of fr(s), u ≡ ua(s), ub(s)s∈S the set of temporaryutility levels of each worker in each region, and n ≡ fa(s), fb(s)s∈S is the worker distribution betweenthe two regions. We assume that the function Φ(·|u, n) is a smooth functional of u and n, and satisÞesthe following condition:24

Φ(s|u, n) ≥ 0 if and only if (ua(s)− ub(s))fa(s)fb(s) ≥ 0. (14)

24In our setting, it can be veriÞed that the temporary utility level of each skilled worker smoothly changes given a smallchange in the distribution of workers between the two regions. Here, we further assume that this smooth response of theutility levels in turn translates into that of migration rates. The concept of the smoothness of a functional is based on theso-called Volterra [40] derivative.

10

The implication of condition (14) for the adjustment process (13) is obvious: the region where workers

can attain higher utility is more attractive, under the constraint concerning the population distribution

given by (3). The migration continues until nobody has an incentive to relocate.

The above considerations lead to the following deÞnition of the (long-run) equilibrium: an equilibrium

is a state of the economy where (i) all the conditions for the temporary equilibrium are satisÞed; (ii) all

workers of the same skill attain the same utility level irrespective of their location. So, an equilibrium is

a temporary equilibrium that satisÞes also the condition that úfr(s) = 0 for all s and r.

Now, how will the equilibrium distribution of workers across regions look like? In searching for

equilibria we perform the following thought experiment. Starting from an arbitrary temporary equilibrium

we ask whether a worker-seller of any skill level s located in region a has an incentive to relocate to region

b.25 She is willing to do so provided the utility level in region b is at least as high as what she is enjoying

in region a. Denote by u(s) the relative standard of living in region b (i.e., u(s) ≡ ub(s)/ua(s)) of a workerwith skill level s. By applying (10) to (12), and using (7) and (11), we obtain the following expression:

u(s) =1

P

sX + (s− c)τσ−1

s+ (s− c)τσ−1X. (15)

where P is the relative cost of living index in region b given by

P ≡ Pb/Pa =·sa + (sb − cnb/bs)τσ−1

sb + (sa − cna/bs)τσ−1

¸1/(σ−1)

, (16)

and X represents the relative local Þrm demand, given by the ratio between the local demand for a Þrm

selling a good of an arbitrary quality level s in region b and that in region a:

X ≡ Xb(s)

Xa(s)=Aa +BbAb +Ba

. (17)

A couple of remarks are in order. First, the relative local Þrm demand X is independent of s.26 Thus, if

the local demand for some variety s is larger in one region, then it is larger also for all the other varieties.

Second, it is important to avoid misinterpretations concerning the meaning of X. If, say, X > 1, then

an arbitrary Þrm has larger local sales if located in region b. This, however, does not mean that, on

aggregate, sales, expenditure, and nominal income are greater in region b. Thus, it is important to bear

in mind that X refers to the relative size of local markets from the viewpoint of the representative Þrm.

A regions attractiveness depends on the individual-speciÞc skill level s, region-speciÞc variables (X

and P ) and the level of transport and communication costs, τ and c. For workers of a given skill level,

what matters are regional variables. Regional differences in market size and the cost of living index are

shaped, in turn, by the allocation of population and skill across locations.

25In the following, we use the terms, relative standard of living, relative utility level, and relative proÞtability, inter-changeably.

26Note that this result does not depend upon the Cobb-Douglas speciÞcation chosen to nest quality and quantity inpreferences.

11

A worker in region a of skill level s has a strict incentive to relocate if and only if u(s) > 1. The

problem is perfectly symmetric when considering a worker with skill s who is instead temporarily located

in region b. In equilibrium, we must have fa(s) > 0 [resp., fb(s) > 0] if and only if u(s) ≤ 1 [resp.,

u(s) ≥ 1].

What matters for the location of worker-sellers in this economy? Recall by (10) that the sales and

earnings of more skilled sellers depends less upon locations. From a consumers point of view, all agents

equally care about the overall price level Pr, which may differ greatly between locations when economic

activities are not equally distributed in space. It is now not difficult to envisage that a highly-skilled

worker-seller will be attracted to the region with a lower overall price level even if the local market for

her product is smaller. This guess will be conÞrmed in the next section.

5 Configurations, existence, and stability of equilibria

What equilibrium conÞgurations are possible in our economy? Are there different ones? If so, under

what conditions will each one be realized? Are these equilibrium conÞgurations mutually exclusive or

not? Are they stable? In this section we address these issues. First, we claim that in our setting, we

have at most the following three equilibrium conÞgurations:

Definition 1 (Equilibrium conÞgurations) (i) Dispersed equilibrium: the most skilled worker in each

region is more skilled than the least skilled worker in the other region; (ii) Concentrated equilibrium: all

mobile workers locate in one region;(iii) Segmented equilibrium: the most skilled worker in one region is

equally skilled or less skilled than the least skilled worker in the other region.

Dispersed equilibria include all the possible equilibria in which both regions have workers of each skill

level. Graphically, they are located in the interior of the area delimited by the Lorenz curves (areaM in

Figure 2). A special case of the dispersed equilibrium is the symmetric equilibrium (at point E in Figure

2), where the aggregate skill and population are equal (though the skill distribution may differ) across

regions. Segmented equilibria are located along the Lorenz curves. The concentrated equilibrium is at

one of the two corners, O and O0, in the Þgure, and is an extreme case of the segmented equilibrium,

where the threshold skill level is out of range [s, s]. Our model shares the possibility of the symmetric

and concentrated equilibria with the standard core-periphery model. However, as we will see below,

segmentation is the most typical stable equilibrium in our model.

In this section, we derive conditions for each equilibrium conÞguration deÞned in DeÞnition 1. We also

analyze the stability of the dispersed and concentrated equilibria, while that of the segmented equilibria

is left for the global stability analyses presented in Section 6.

12

5.1 Segmented equilibrium

We start from the segmented equilibrium, since it is easy to ascertain that in our setup this conÞguration

is the rule. In these equilibria, workers with similar skill endowments tend to cluster in the same location.

By (15), it can be shown that the relative local Þrm demand directly affects the relative proÞt. Taking

any s ∈ (s, s) we have

u(s) ≥ u(s0) if and only if X ≤ 1 for s0 < s. (18)

How do we interpret these relations? Consider X < 1, so that region a offers a larger local market to each

Þrm. Recall that a smaller value of u(·) makes region a more attractive. From (18) we see that region

a is less attractive for high-skilled workers (or, if we prefer, for Þrms selling high-quality goods). This

means that being located in a region offering a smaller local market for their goods is comparatively less

disadvantageous (or more advantageous) for the more skilled. The explanation is simple. Communication

costs fall more heavily on low-quality suppliers. Hence, those sellers that supply goods of higher quality

are relatively footloose, since their sales can penetrate distant markets easily. We can reverse the argument

for the case of X > 1. When X = 1, the relative proÞtability coincides for all Þrms. We can therefore

state the following lemma:

Lemma 1 Whenever local sales of any given Þrm differ across locations (i.e., X 6= 1) the equilibrium

can only be either concentrated or segmented. Moreover, in a segmented equilibrium, all the workers with

sufficiently high skills are located where their local sales are smaller.

Lemma 1 says that in a segmented equilibrium, less skilled workers tend to cluster in a region where

their local sales are abundant but where locally supplied goods are of relatively low quality. Conversely,

workers with sufficiently high skills Þnd it convenient to locate where the local market for their good

is smaller, but where locally provided goods are of relatively high quality. This result is shaped by the

interaction between transport and communication costs and skill heterogeneity. Since for the less skilled

it is more difficult to penetrate distant markets due to the loss of quality in transit, they are more tied to

the local market. The more skilled are instead relatively footloose and are at ease in locating themselves

where locally supplied goods are of high quality and the true cost of living index is lower. We will see in

the subsequent sections that the low price level in the high-skill region will indeed more than compensate

for the disadvantage of a small local market.

Due to the possible multiplicity of equilibria, the stability of the segmented equilibria is not apparent.

This is investigated in Section 6.

13

5.2 Dispersed equilibrium

By Lemma 1, in any dispersed equilibrium we must have X = 1 (equal sales for all Þrms in both regions).

Moreover, we see from (18) that when X = 1, the relative standard of living must be the same for all

agents. A dispersed conÞguration thus requires u(s) = 1 for all s. Finally, we note from (15), that X = 1

and u(s) = 1 also imply that P = 1 at any dispersed equilibrium. As will be clear, the only conÞguration

consistent with X = P = 1 is the symmetric equilibrium, where the mass of population and that of skill

are equal in the two locations. Consider a worker endowed with an arbitrary skill level s. From (15), we

can show that the impact of the local market size and cost of living on the relative standard of living is

described by the following condition:

u(s) < 1 if and only if

½X < 1 and P ≥ 1, orX ≤ 1 and P > 1. (19)

That is, given the same price level [resp., local Þrm market size], the region yielding a larger local market

[resp., lower price level] is more attractive to any worker. We know that, at any dispersed equilibrium, all

workers must be indifferent between the two locations. It follows that the regional allocation of workers is

necessarily indeterminate in this case. However, the values of X and P depend upon how the aggregate

skill and population are shared between the two regions. So, to characterize the dispersed equilibrium

we must Þnd out the particular class of worker distributions which are consistent with X = P = 1.

How does the regional shares of population and skill (na, sa) affect the value of X and P ? It can be

shown from (2) and (17) that its impact on X is described as follows:

X ≥ 1 if and only if na ≥µ1− σ − 1

σ + 1τ1−σ

¶ bsc(sa − 1/2) + 1/2. (20)

This relation indicates Þrst that given an aggregate skill in a region (sa), a larger number of competitors

(na) will make the local Þrm demand less than that of the other region (Xa is smaller than Xb). Next,

consider the (X = 1)-line in the (na, sa)-space. Equation (20) suggests that the line passes through

(1/2,1/2), and rotates anti-clockwise as transport costs increase (τ decreases). It is convenient to have

a glance at Figure 3, in which Diagram (a) [resp., (b)] depicts the (X = 1)-line on the feasible (na, sa)-

domain shown in Figure 2 for the case of low [resp., high] transport costs. The mechanism for this can be

understood easily if we consider the location incentive of worker-sellers when all of them are concentrated

in one region, say region a (which is at point O0 in the diagrams). For each worker-seller, the beneÞt

of leaving region a is that she can enjoy a larger market share in region b where there are no local

competitors, while the cost is that she is moving away from the large market in region a. When transport

costs are low enough (sufficiently large τ), the competitors in region a can easily reach the market in

region b, and hence, the cost dominates, i.e., X < 1 (the slope of the (X = 1)-line is between 0 and 1 as

in Diagram (a)). If instead transport costs are high enough (sufficiently small τ), so that the market is

spatially segmented, the beneÞt dominates, hence X > 1 (the slope of the (X = 1)-line is either negative

14

or greater than 1 as in Diagram (b)). It is to be noted that the presence of communication costs softens

the impact of the spatial distribution of skill on the relative local Þrm demand. When c is very high,

what really matters for the local sales of each worker-seller is the number of local competitors, not their

skills. Next, by (2) and (16), we have

P ≥ 1 if and only if na ≥ (1− τ1−σ)bsc(sa − 1/2) + 1/2. (21)

The interpretation of (21) is straightforward. Given the regional allocation of aggregate skill (sa), a larger

number of sellers in a (na) lowers the cost of living there (Pa decreases relatively to Pb) due to consumers

love for variety. A larger number of local sellers is associated indeed with more varieties available locally.

On the other hand, given the regional allocation of population, a larger aggregate skill in a region implies

a greater average quality of goods, and hence, a smaller cost of living. It follows that the slope of the

(P = 1)-line is always negative in the (na, sa)-space (refer to Figure 3). The presence of communication

costs again softens the impact of the spatial distribution of skill on the relative cost of living: given a

larger value of c, the relative price level more crucially depends on the size of locally available product

variety rather than their quality.

From (20) and (21), the relation between the (X = 1)- and the (P = 1)-lines can be described as

follows. Both lines rotate anti-clockwise in the (na, sa)-space as τ decreases, and converge to sa = 1/2

as τ approaches 0. It is apparent that the two lines never coincide except at the symmetric equilibrium

(1/2,1/2). This proves that the symmetric equilibrium is the only dispersed equilibrium.

Is the symmetric equilibrium stable? The answer can be a positive one only if we can show that

the symmetric equilibrium is restored after any possible perturbation to the distribution of workers. It

is not difficult to show that this can never be the case. Consider a perturbation such that sa > 1/2.

Then, along the (X = 1)-line we must have P > 1, so that u(s) < 1 for all s. This means that in any

arbitrarily small neighborhood of the symmetric equilibrium there always exists a perturbation to the

regional distribution of workers (along the (X = 1)-line) which leads to a self-reinforcing agglomeration

process towards region a. Symmetrically, we can always Þnd a perturbation that induces sa < 1/2 and

that leads to a self-reinforcing agglomeration towards region b. Thus, we have the following proposition:

Proposition 1 The symmetric equilibrium always exists, and is unstable. Moreover, it is the only dis-

persed equilibrium.

In spite of the fact that the perfectly symmetric regional distribution of workers always induces

an equilibrium, this equilibrium cannot be a stable one. This is a major point of departure of our

analysis from the standard core-periphery model (and its existing variants), since there the symmetric

equilibrium can instead be stable. The instability arising in our model is generated by the interaction

of skill heterogeneity with communication costs. It is to be noted that the instability result persists

15

even if the skill heterogeneity of workers is marginal, and as long as communication costs are positive,

even if very close to zero. This indicates that the simultaneous presence of worker heterogeneity and

communication cost is a necessary and sufficient condition for the instability result.

This observation leads to some remarks of possible interest. The Þrst concerns the robustness of

the results arising under the conventional framework of the core-periphery model. Agent heterogeneity

reveals a major source of instability, that has so far received insufficient attention. Second, the presence of

external economies related to human capital may not be necessary to predict the inevitability of regional

differences in endowments and income. Pecuniary externalities are enough to do the job. Finally, we

remark on the role of communication costs. It is becoming a popular view that falling communication

costs due to the advancements in telecommunication technologies will in the end eliminate the relevance

of physical space for economic activities. Production will become footloose, and big, crowded cities will

be replaced by telematic villages (see Glaeser [24] for a discussion). Our model predicts that this view

is correct only if taken as a hyperbolic statement. To recover the stability of dispersed equilibria, and

hence, the regional convergence, communication costs must totally disappear: lowering them will not be

sufficient.

5.3 Concentrated equilibrium

This is another polar case that emerges in the standard core-periphery models, where all mobile workers

concentrate in one region. In our model, however, the concentrated equilibrium can also be interpreted

as a special case of the segmented equilibrium. In this particular case, although workers with different

skills tend to have different location incentives, all worker-sellers prefer to be located in the same region.

In analyzing existence and stability of the concentrated equilibrium, it is convenient to answer Þrst

the following question: Who are the workers that Þrst leave the concentration when it ceases to be an

equilibrium? An easy argument shows that those workers that have the highest incentive to move away

from a concentrated conÞguration are the least skilled. Suppose all mobile agents are located in one

region, say a. Then, the cost of living is higher in region b, where all goods must be transported from

region a (i.e., P > 1). By (15) this means that region b must offer larger earnings to attract workers

(i.e., X > 1). We know by (18) that in this case we must have u(s) < u(s0) for s0 < s , namely, that

relocating to region b is more advantageous for less skilled workers. It follows that if a worker has an

incentive to move to region b, then all the less skilled workers want to do the same. Hence, we conclude

that whenever the concentration breaks, the least skilled worker is the Þrst one to relocate.

Next, under what condition is the concentration an equilibrium? To simplify the analysis, we consider

the situation in which communication costs alone (i.e., when τ = 1) are not enough to break the concen-

tration. From the discussion above, when all workers are located in region a, a sufficient condition for the

concentrated equilibrium is X < 1, i.e., point O0 is at the left of the (X = 1)-line in the (na, sa)-space as

16

in Figure 3(a). By (20), this is guaranteed if τ ≥ bτ , wherebτ ≡ µ

σ − 11 + σ

1

1− c/bs¶ 1

σ−1

. (22)

So, the desired situation is realized under bτ ≤ 1. Equation (22) implies that this condition is equivalentto bs ≥ σ+1

2 c. Under Assumption 1, it can be written as

Assumption 2 σ+12 s ≤ bs.

In the rest of the paper, we maintain this assumption. In this context, when transport costs are low,

the concentration is an equilibrium. As transport costs increase, workers become less footloose, and seek

the region with lower competition. As a result, the concentration tends to break. However, if the goods

are more differentiated (σ is smaller), transport costs matter less, which in turn implies lower location

dependence of income for all workers. In fact, if the goods are too differentiated (σ < 2), once all workers

agglomerate in one region, no one would move away given any level of transport costs.27

The following proposition summarizes the results obtained [see Appendix A.1 for a proof of statement

(ii)]:28

Proposition 2 (i) When all skilled workers are concentrated in one location, if a worker with skill s0

has an incentive to move to the other region, then all workers with s ≤ s0 want to move as well. (ii) Forσ ≤ 2, the concentration is always an equilibrium, while for σ > 2, the concentration is an equilibrium if

and only if τ c ≤ τ ≤ 1, where τ c ∈ (0, bτ) is the unique solution to the equation: u(s = s; τ) = 1.Notice that if the concentration is an equilibrium, it is by deÞnition stable. Unlike the symmetric

equilibrium, our model shares with the standard core-periphery models the basic ingredients of the

equilibrium condition for the concentration. Provided that the degree of scale economies is not too high,

the concentration is viable when the transport costs are sufficiently low. In our setting, however, we need

another condition for this result, namely, that the level of communication costs is not too high.

6 Global dynamics and inevitable regional inequality

In this section, we derive the global behavior of the economy. From the results in the previous section,

we know that stable equilibrium conÞgurations are either concentrated or segmented, i.e., on the Lorenz

curves. Given the smoothness of the migration process, any worker distribution (except those yielding

27This corresponds to the so-called black-hole condition in the core-periphery model (see Fujita, Krugman, and Venables[21, Sec. 4.6]).

28Statement (i) of the proposition holds without Assumption 2.

17

a symmetric equilibrium) will inevitably converge to an equilibrium exhibiting regional inequality. In

particular, we can show that a cycle is impossible in our economy.

Below, we illustrate the global dynamics of the economy under the assumption that σ > 2, so that

the concentration is not always an equilibrium (Proposition 2 (ii)). The results concerning the possible

equilibrium conÞgurations and their stability are summarized in the following proposition [see Appendix

A.2 for the proof].

Proposition 3 (i) If bτ < τ < 1, then the concentrated equilibria exist, and there is no segmented

equilibrium. (ii) If τc < τ < bτ , then the concentrated equilibria exist, and there may also exist one pairof segmented equilibria (one stable equilibrium and one unstable) with X < 1 and one pair with X > 1.

(iii) If 0 < τ < τ c, then the concentrated conÞguration is not an equilibrium, and there exist at least one

stable segmented equilibrium with X < 1 and one with X > 1. (iv) There is no cycle.

Figure 3 conveys the basic message of the proposition. Diagram (a) depicts the case of low transport

cost (bτ < τ), while Diagram (b) depicts the case of high transport cost (τ < τc), where the gray area

is the same feasible domain M shown in Figure 2.29 The arrows in the diagrams indicate the typical

direction of adjustments.

Figure 3

In Diagram (a), it can be seen by (19) that u(s) < 1 for all s in area A0EB, so that all dynamic

trajectories starting in the area will eventually reach the concentrated equilibrium at O0. Similarly,

those starting in area AEB0 will reach O. The reason is straightforward. In these areas, both skill and

population are relatively concentrated in one region. In this region, the cost of living is lower, since goods

of higher quality and a wider variety are available in the more populated region. Moreover, since the

competition is global when transport costs are low, locating in the less populated region (i.e., with fewer

local competitors) will not improve sellers market size much. Thus, the cost of living is the dominant

location determinant in this case. In areas AEB and A0EB0, the adjustment direction at each point is

instead ambiguous. Since we have X < 1 and P < 1 in area AEB, despite region a having a larger market

(which is good for Þrms), it has a higher cost of living (which is bad for consumers). The ambiguity of the

migration direction in area A0EB0 can be explained similarly. It can be shown, however, that in Diagram

(a) any dynamic trajectory starting in these areas will eventually either reach the symmetric equilibrium

E, or enter area AEB0 or A0EB (refer to Lemmas 2 and 3 in Appendix A.2). Note in particular that

there are no dispersed equilibria in the interior of areas AEB and A0EB0 (Proposition 1). Also, by (20),

we have X < 1 at any point along OB. Then, by Lemma 1, any segmented conÞguration along OB

29We omit the diagram for the case of intermediate transport costs (τc ≤ τ ≤ bτ), since it adds no signiÞcant results. Forthis case, refer to Figure 5(c,c) in Appendix A.2.

18

cannot be an equilibrium, since in this conÞguration the less skilled are located in region b, where the

local Þrm demand is smaller. By the same argument, there is no equilibrium along O0B0. Hence, any

initial worker distribution in M will eventually end up with a complete concentration in one region.30

In Diagram (b), areas A0EB and AEB0 are characterized by the same migration pattern as those

in Diagram (a). Thus, dynamic trajectories starting in these areas eventually enter areas A0EB0 and

AEB, respectively. Under high transport costs (τ < τc), unlike the case of Diagram (a), the intensity

of competition differs signiÞcantly across regions when skill and population are relatively concentrated

in one region. In particular, when all sellers are located in one region (O or O0), the least skilled would

always want to deviate from the concentration to escape from tough competition (Proposition 2 (i)). Also,

as explained above, a segmented conÞguration is not an equilibrium along OB and O0B0. It follows that

the only possible stable equilibria are the segmented ones along OA and O0A0. The arrows in Diagram

(b) are for simplicity drawn for the case in which there is a unique segmented equilibrium on OA and on

O0A0.

In the case of intermediate transport costs (τ c ≤ τ ≤ bτ), the possible equilibrium conÞgurations are

either the same as those in Diagram (a), or a situation in between Diagrams (a) and (b) may happen, where

both the concentrated and segmented equilibria coexist, as statement (ii) in the proposition indicates.

7 Discussion

We saw that regional inequality is inevitable in our setting. This raises a number of questions. Where

should we expect the most skilled agents to be located? What is the role of transport and communication

costs in shaping the spatial distribution of economic activities? What are the links between skill inequality

and agglomeration? The subsections below discuss these issues.

7.1 Skills and location patterns

Where do the most talented go? Our analysis above indicates that in any stable equilibrium, the aggregate

skill in the region populated by the highly-skilled workers must be at least a half of the total, i.e., bs/2. Thiscan be seen from the fact that the share of aggregate skill in region a at point A0 the intersection of the

(X = 1)-line and the upper Lorenz curve can never be smaller than 1/2 by (20). As for the population

share in this high-skill region, things are less clear-cut. Figure 3(b) shows that in all segmented equilibria

the region inhabited by the high skilled must have a minimum population size (i.e., when the skilled go

to region a, na is necessarily bigger than the value found at the intersection of the (X = 1)-line and

the upper Lorenz curve). This minimum population size, however, could be below 1/2, provided the

(X = 1)-line is negatively sloped (i.e., transport costs are high).31

30Except the measure zero area which is contained in a stable manifold leading to E.31Recall that all the segmented equilibria on the upper and lower Lorenz curves are respectively along the O0A0 and along

OA segments in Figure 3(b).

19

We can say the following. The most skilled are always found in wealthier locations, where aggregate

human capital, and aggregate income (and welfare) is higher.32 Wealthier locations may end up being

less populated, because these are inhabited by the privileged few who are skilled and rich.

How can we explain that the highly-skilled are to be found in wealthier locations? The reason is tied

to the different impact of communication costs on sellers of different skills. Since the more skilled are

always closer to the global market (less dependent upon local markets), they are not as sensitive as the

less skilled to being located where their local individual demand is low. In other words, the local Þrm

demand is small in rich thriving areas. Fierce competition is the price to pay for being located where the

best are: the local total demand must be shared among the top sellers, and the share accruing to each

is necessarily small. However, note that the cost of living index is lower in the high-skill region than in

the low-skill region, since a wider range of high-quality goods are available there. This low price index

compensates for the disadvantages of a limited local market, yielding a high standard of living in the

high-skill region.33 We summarize these remarks in the next corollary of Proposition 3.

Corollary 1 (i) In any stable equilibrium, one region has higher aggregate skill and income, as well as

a higher average utility level than the other. (ii) Workers at the top of the skill ladder are always located

where aggregate skill is higher.

7.2 The role of transport and communication costs

Distance matters in our model because of both transport and communications costs. As for transport

costs, we see from Proposition 3 that whenever transport costs are sufficiently low (i.e., τ approaches 1)

the equilibrium conÞguration is necessarily concentrated. The maximum extent of agglomeration occurs.

This is a basic result of the standard core-periphery model. As transport costs rise, however, we cannot

hope for a symmetric conÞguration in our model. The conÞguration will be segmented, with one high-

skill region and one with low skills. We can say more. If transport costs are not too high, the high-skill

region will be the one with the larger population. This is easily understood recalling that, by (20), the

(X = 1)-line is positively sloped if τ > (σ−1σ+1)

1/(σ−1) (refer to Figure 3(a)). By the previous arguments,

any segmented equilibrium must exhibit larger population in the high-skill region in this case. Recall also

that the (X = 1)-line rotates clockwise as transport costs fall. So, we see that a reduction in transport

costs still tends to raise the extent of agglomeration as in the standard core-periphery model. As τ

becomes close to 1, most of the mobile agents must be located in the high-skill region.

32Denote by I = Ib/Ia relative nominal incomes. After some algebra it is shown that that I > 1⇔ sa <1−c/bs2−c/bs+ c/bs

2−c/bsna.

It is easily checked that this condition is satisÞed if and only if the most skilled are located in b.33A straightforward corollary of this result is that immobile workers are necessarily better-off in the high-skill location.

Their nominal income is the same in both locations, but the real income is higher in the high-skill region because of a lowercost-of-living index.

20

What about communication costs? It is possible to show that a fall of communication costs may have

very different effects on the equilibrium conÞguration depending on the level of transport costs.

Consider the case where transport costs are low (τ > (σ−1σ+1 )

1/(σ−1)). Then, communication costs

are dominating in the overall cost for transit. Then given a decrease in communication costs, even low-

skilled worker-sellers become footloose. We see that in this context the agglomeration of human capital

is necessarily associated with that of population. This can be seen from the fact that the range of

possible segmented equilibria (OA and O0A0 on the Lorenz curves) shrinks after a fall in communication

costs (refer to (20)). On the other hand, if transport costs are high (τ < (σ−1σ+1)

1/(σ−1)), through a

symmetric mechanism, even highly-skilled workers are tied to the local market, and seek the region with

less competition. As a result, the agglomeration of human capital is not likely to be associated with

that of population as communication costs fall. These Þndings are summarized in the next corollary to

Proposition 3:

Corollary 2 (i) When transport costs are low (τ > (σ−1σ+1 )

1/(σ−1)) the region with higher aggregate skill

has also a larger population. (ii) When τ > (σ−1σ+1 )

1/(σ−1) [resp., τ < (σ−1σ+1)

1/(σ−1)], the minimum share

of population in the high-skill region rises [resp. falls] as communication costs fall.

The above results may shed light on some real-world phenomena. It may help to explain for instance

the empirical fact that high concentrations of human capital have been associated with those of population

in recent decades characterized by substantially decreasing communication costs and a fairly low level

of transport costs (Glaeser et. al. [25]; Black and Henderson [6]). The model also helps in qualifying

predictions concerning the impact of reduced communication costs on spatial agglomeration. In general,

the analysis gives a further argument in favor of those that are skeptical about an inevitable dispersion

of economic activities as a result of lower costs for communications: segmentation or concentration will

remain the only stable outcome even with very small communication costs.34 However, communication

costs may matter for the extent of agglomeration because they affect the relation between the equilibrium

location of skills and population.

7.3 Agglomeration and interpersonal inequality

How interpersonal (skill and then income) inequality relates to regional inequality? Due to the per-

vasiveness of equilibrium multiplicity a general assessment concerning the links between inequality and

agglomeration is beyond the scope of our analysis. Formally, we cannot pin down the location and

number of the segmented equilibria along OA and O0A0 on the Lorenz curves in Figure 3(b). Given

the relatively simple model structure, however, once the functional form of the skill distribution among

workers is speciÞed, we can numerically explore the behavior of the model fairly exhaustively over the

34See, for instance, Gaspar and Glaeser [23], for an argument in favor of greater need of face-to-face contact, and agentproximity in the presence of lower communication costs.

21

possible parameter range. For simplicity, we assume a uniform distribution of population over the skill

range [s, s].35 The key parameters are the aggregate skill in the economy, bs, the degree of interpersonalinequality (as measured by the Gini index), and the parameters for transport and communication cost,

τ and c.36

Regarding the number of equilibria, our simulation results indicate that there are at most two stable

equilibria along each of the OA and the O0A0 segments on the Lorenz curves.37 Figures 4 and 5 present

typical numerical examples showing the relation between the degree of interpersonal skill inequality and

that of population and skill agglomeration. We consider two ways to increase the Gini index. The Þrst is

to increase the aggregate skill, bs, along with the highest skill level, s, while keeping the lowest skill level,s, Þxed. The second is to expand the skill range [s, s], while keeping the aggregate (and average) skill level

constant. In the Þrst case a higher value of the Gini index is necessarily associated with a higher value of

bs, while in the second case a higher value for the Gini index corresponds to a mean preserving spread inthe skill distribution obtained by raising s and lowering s by the same amount. Figure 4 describes how

the equilibrium conÞgurations behave when inequality changes in the Þrst way, while Figure 5 depicts the

second case. The horizontal axis in each diagram in Figure 4 [resp., 5] measures the value of bs [resp., s],which, as explained above, represents the degree of interpersonal inequality in this case. The vertical axis

in Diagram (a) [resp., (b)] in each Þgure measures the share of skilled population [resp., the aggregate

skill share] in the high-skill region (which is assumed to be region a). In each diagram, different curves

correspond to different values of τ , while other parameter values are indicated in the Þgures. In both

diagrams in Figure 4, for ease of understanding, the unstable equilibria are drawn by dotted curves when

there are two stable equilibria for given parameter values.38

Figure 4

Consider Þrst Figure 4. There, the minimum skill level is Þxed at s = 1, while the aggregate skill

bs is varied from 1.0 to 25.0.39 Diagram (a) indicates that if transport costs are sufficiently low (τ =

0.8513, τ = 0.8525), then the population share of the high-skill region is likely to rise as the interpersonal

inequality increases (i.e., bs increases). The relation between the extent of agglomeration and that of35One of the important implications of this setting is that when interpersonal inequality increases, the boundaries of

feasible domainM (refer to Figure 2) expands monotonically, i.e., each point on the upper [resp., lower] Lorenz curve movesupward [resp., downward].

36The effect of the substitution elasticity σ can be deduced straightforwardly from the result of the standard core-peripherymodel: a smaller σ (higher degree of differentiation) is associated with higher population (i.e., variety) concentration (inour context, in the high-skill region). It is because the sales is less dependent on transport cost when the goods are moredifferentiated from one another.

37In fact, these equilibria bifurcate from the concentrated and symmetric equilibria of the core-periphery model. As sand c approach s and 0 respectively, these segmented equilibria converge to the equilibria of the core-periphery model.

38The unstable equilibria are omitted in Figure 5. But, whenever there are two stable equilibria, there is one unstableequilibrium between them.

39All the numerical examples in Figure 4 start from the symmetric equilibrium at bs = s = 1.0, where there is nointerpersonal inequality. For τ close to 1, our simulation indicates that the possible stable equilibrium is instead only theconcentration when bs =s under Assumption 2, and that the concentration continues to be a stable equilibrium for all bs >s.

22

inequality seems however highly nonlinear. We see indeed that the share of population in the high-skill

region may actually fall when the interpersonal inequality is small. Diagram (b) indicates that when the

transport costs are low, the aggregate skill share in the high-skill region is likely to rise as the interpersonal

inequality increases (τ = 0.8513, τ = 0.8525). Recall that the segmented equilibrium is located along

the segment OA or O0A0 of the Lorenz curves shown in Figure 3(b). It follows that if interpersonal

inequality increases in such a way that the feasible domain M expands monotonically at each point on

the Lorenz curves (as in the case of our simulation setting), then we can unambiguously say that skill

agglomeration will go up when the rise in inequality induces more population agglomeration; though

the impact is ambiguous when the population agglomeration instead decreases. The simulation results

show that for small transport costs (τ = 0.8513, τ = 0.8525), the skill agglomeration rises even when

the population agglomeration is falling, while it may decrease with the share of population in the region

when the transport costs are high (τ = 0.8, τ = 0.85). From these results we can argue that when higher

interpersonal inequality is associated with a growth in the aggregate human capital and transport costs

are low, skill and population tend to agglomerate in the same region as this inequality rises.40

We can interpret this behavior of the equilibrium conÞgurations as follows. When an increase in

interpersonal inequality is associated with the growth of overall human capital, the fraction of skilled

workers whose earnings are less dependent on location rises with the degree of interpersonal inequality.

As a result, the population would tend to move into the high-skill region where the cost of living is lower.

Hence, the agglomeration of skill and that of population tend to coincide when inequality is sufficiently

large. Not surprisingly, greater dominance of communication costs over transport costs, and hence, less

dependence of overall costs for transit on the sales value, will enhance this tendency of co-agglomeration

of population and skill. These results are consistent with the continuing urbanization over the centuries

in the world economy where income, human capital, and often inequality have been rising (Bairoch [2]).

Figure 5

In Figure 5, the aggregate skill is Þxed at bs = 25.0, and the highest skill s is varied from 25.0 (namely,the value of s) to 49.0, at which s = 1.41 A glance at the diagrams is sufficient to conclude that when

higher interpersonal inequality is generated by a mean preserving spread in the skill distribution, the

agglomeration of skill becomes less correlated with that of population as inequality rises. The underlying

mechanism is simple. A greater interpersonal inequality under a given aggregate skill in the economy

implies a greater concentration of skill to a fewer number of workers. Thus, the earnings of most of the

workers heavily depend on the size of the local market, which depends in turn on the degree of local

40From the simulation results in Figure 4 we also notice that the range of interpersonal inequality in which the regionwith a larger population and that with larger human capital do not match is smaller when transport costs are low.

41The equilibria at bs = 25.0 in Figure 4 correspond to those at s = 49.0 in Figure 5.23

competition. Consequently, one may observe a large population also in a region with a small aggregate

skill endowment. This result may help to interpret some stylized facts that are observed in cross country

comparisons. For instance, if we compare the urban structure in countries with similar income per capita

such as the US and Japan, cities tend on average to be more populated in Japan, where the extent of

interpersonal inequality is smaller.42

8 Concluding remarks

Economic agents are not all equal. This is a basic fact of life. Some workers, sellers, entrepreneurs are

more skillful, brilliant, or simply luckier than others. In a world of unequal abilities and fortunes, we

necessarily observe rich and poor places: wealth and human capital are not evenly represented across

towns, regions or states. This paper addresses these points formally. Our economy is populated by sellers

that differ in their skills, thus performing differently in the marketplace. We show that modifying the

well-known core-periphery model of location in such a way that agent heterogeneity is allowed has major

analytical implications. The sustainability of a symmetric location pattern, which is a common feature

of the existing new economic geography models, breaks. Regional inequality is inevitable. Most skilled

agents are attracted by wealthier locations, those where human capital and wealth are more abundant.

42In 1990, the largest three metropolitan areas account for 14% of the total population in the US. The correspondingnumbers are 21%, 21% and 31% in France, Italy, and Japan, respectively (United Nations [39]).

24

A Appendix

A.1 Proof of Proposition 2(ii)

Without loss of generality, assume all workers are located in region a (sa = na = 1; sb = nb = 0). In this

case, the expressions for the relative price index (16) and the relative local Þrm demand (17) boils down

to P = (1− c/bs) 11−σ /τ and X = σ−1

σ+1τ1−σ1−c/bs , respectively. By substituting these expressions into (15), we

obtain

u(s) = (1− c/bs) 1σ−1

(σ − 1)τ2−σ + (σ + 1)(1− c/s)(1− c/bs)τσ(σ + 1)(1− c/bs) + (σ − 1)(1− c/s) . (23)

From the investigation of (23), we can show the following. If σ ≤ 2, then u(·) = 0 at τ = 0, and u(·)is increasing in τ > 0. If σ > 2, then u(·) approaches ∞ as τ approaches 0, and as τ increases from 0,

it Þrst decreases, then increases (in particular, it is convex for τ > 0). Statement (ii) follows from this

result together with statement (i). Q.E.D.

A.2 Proof of Proposition 3

We can classify the global dynamics of regional population and aggregate skill, (na, sa), into four typical

patterns: cases of (a) low transport cost (bτ < τ < 1), (b) high transport cost (0 < τ < τ c), and (c)

intermediate transport cost (τ c < τ < bτ) and high skill inequality (large M), (c) intermediate transportcost (τc < τ < bτ) and low skill inequality (small M) .These patterns are mutually exclusive, and also exhaustive as far as the possible types of equilibria

are concerned. That is, they are not exhaustive only in cases (b,c) where the actual number of segmented

equilibria may be different depending on parameter values.

Depending on the relative home market size X and the relative price index P , the feasible domainM

can be decomposed as follows. Given a contiguous domain N ⊂ M , we say that it is of type Ωa [resp.,Ωb] if u(s) < 1 [resp., u(s) > 1] for all s at each point in N ; type Ωc [resp., Ωd] if there exists z ∈ (s, s)such that u(z) = 1, u(s) > 1 [resp., u(s) < 1] for all s > z, and u(s) < 1 [resp., u(s) > 1 ] for all s < z

(i.e., X < 1 [resp., X > 1] by (20)) at each point in N .

Regarding the migration pattern in each domain type, we can show the following:

Lemma 2 (Direction of local adjustments) At each point in a domain of type Ωa,Ωb,Ωc, and Ωd, the

direction of local adjustment ( úna, úsa) under the dynamics (13) is in, respectively, the Þrst quadrant (Þgure

2), the third quadrant, the domain below the 45 line through the origin, and the domain above the 45

line through the origin, where úna and úsa are the time derivatives of na and sa.43

Proof : Since in any domain of type Ωa [resp., Ωb], we have u(s) < 1 [resp., u(s) > 1] for all s, the

adjustment at each point in the domain takes place so that both the population and the aggregate skill

in region a [resp., b] increases.

43In fact, we can narrow down the adjustment direction in type Ωa and type Ωb domains even more. Namely, the steepestincrement [resp., decrement] of the aggregate skill in each region is s [resp., s]per unit increase in the population. So, theadjustment direction is restricted between the vectors (1, sas) and (1, sas) [resp., (−1,−sas) and (−1,−sas)] in the area oftype Ωa [resp., Ωb].

25

Next, consider any domain of type Ωc. Since X < 1 at each point in the domain, we know that

u(s) > u(s0) if s > s0, which implies that workers with skill level s < z want to locate in region a, while

those with s > z in region b. Then, the direction of the adjustment vector ( úna, úsa) at each point in the

domain has an acute angle to the vector (1, zo) normal to (1, z) (refer to Diagram (a) in Figure 6). Note

that for 0 <s< s < ∞, vector (1, zo) is directed to the southeast (i.e., in the forth quadrant) for anythreshold skill level, z ∈ (s, s). It follows that the adjustment at each point projected on the direction ofvector (1,-1) is positive. In other words, the direction of adjustment at each point in the domain is below

45 line passing the point. (that is, toward either the east-northeast, southeast, or south-southwest). It

can be shown in a similar manner that the direction of the adjustment at each point in any domain of

type Ωd is above the 45 line passing the point (that is, toward either the north-northeast, northwest, or

west-southwest) (refer to Diagram (c) in Figure 6). Q.E.D.

Figure 6

Note that if O [resp., O0] is contained in a domain of type Ωa [resp., Ωb], then it is a stable equilibrium.

Note also that if the segmented equilibria exist, they must be on the part of the upper [resp., lower] Lorenz

curve which is contained in a domain of type Ωd [resp., Ωc].

Using Propositions 1, 3, Lemmas 1 and 2, we can decompose domain M into types introduced above.

The result is shown in Figure 7. Each diagram in the Þgure depicts the same Lorenz curves we have in

Figure 5. Diagrams (a,b) in Figure 5 correspond to Diagrams (a,b) in Figure 7. Figure 7 also shows the

case of intermediate transport cost (τc < τ < bτ) in Diagrams (c,c).Figure 7

Lemma 3 (Decomposition of feasible domain M) (i) Area A0EB [resp., AEB0] (including the boundary)

is of type Ωa [resp., Ωb]. (ii) Area AEB [resp., A0EB0] consists either of a type Ωc [resp., Ωd] domain

entirely, or of domains of type Ωa and/or type Ωb isolated by a type Ωc domain. (iii) The line segments

A0E and BE [resp., AE and B0E] are contained in a type Ωa [resp, Ωb] domain except for the end point

E. (iv) The neighborhood of O0 [resp., O] (of positive measure) is either of type Ωb or of type Ωd [resp.,

Ωa or Ωc] if τ < τc.

Proof : By (19), we know that u(s) < 1 [resp. u(s) > 1] for all s at each point in area A0EB [resp., AEB0]

(including the boundary) in all the diagrams in Figure 7. Hence, this area is of type Ωa [resp., Ωb], and

statement (i) is proved.

By (20), in Diagrams (a,c,c), in the area above [resp., below] the (X = 1)-line (AA0), we have the

area X < 1 [resp., X > 1] at any point. It follows that u(s) > u(s0) [resp., u(s) < u(s0)] if s > s0 at

any point in this area. On the other hand, by (20), again, in Diagram (b), in the area above [resp.,

below] the (X = 1)-line (AA0), we have X > 1 [resp., X < 1] at any point. It follows that u(s) < u(s0)

[resp., u(s) > u(s0)] if s > s0 at any point in this area. Note that within the area of X < 1 and that of

X > 1, a type Ωa and a type Ωb domains cannot have a non-empty intersection except for the symmetric

equilibrium, E.

26

Let us Þrst deduce types of the rest of domain M in Diagram (a). The continuity of u(·) in na andsa, and the fact that a type Ωa and a type Ωb domains cannot have common points except E imply

that we can Þnd a continuous curve contained in area AEB connecting E, and a point, say, C, on the

segment AB of the upper Lorenz curve, such that u(s) = 1 and u(s) < 1 for all s < s along curve CE,

and another continuous curve in area AEC, connecting E and a point, say, D, on the segment AC of

the upper Lorenz curve, such that u(s) = 1 and u(s) > 1 for all s > s along DE.44 Moreover, curves CE

and DE can be chosen so that the entire area BEC is of type Ωa, and area AED is of type Ωb.45

By the symmetry of the adjustment dynamics (13) with respect to the symmetric equilibrium, E, we

can also Þnd at the symmetric location a pair of curves C 0E and D0E connecting E and the segment

A0B0 of the lower Lorenz curve. We have that u(s) = 1 and u(s) < 1 for all s > s along curve D0E, while

u(s) = 1 and u(s) > 1 for all s < s along C0E, and that area A0ED0 is of type Ωa, and area B0EC0 is of

type Ωb.

Since u(s) > u(s0) [resp., u(s) < u(s0)] if s > s0, at each point in area CED [resp., C0ED0], and since

a domain of type Ωa and that of type Ωb cannot have common points except E in an area with a common

sign of X − 1, it follows that area CED [resp., C0ED0] must consist either of a domain of type Ωc [resp.,

Ωd] entirely, or of isolated domains of type Ωa and/or type Ωb surrounded by a domain of type Ωc [resp.,

Ωd].

In a similar manner, in Diagrams (b,c,c), we can Þnd corresponding points C and D [resp., C0 and

D0] on the segment AOB [resp., A0O0B0] of the Lorenz curve, where curves CE,DE,C0E, and D0E have

the same meaning as those in Diagram (a). Area CED [resp., C 0ED0] in Diagram (c) and area CED

[resp., C0ED0] in Diagram (b,c) has the same properties as area CED [resp., C0ED0] in Diagram (a).

Thus, (ii) and (iii) have been proved.

Area CED and C 0ED0 in Diagrams (b,c,c) can be further decomposed. Recall that the concentrated

distribution is an equilibrium if and only if τ c < τ < 1. That is, point O [resp., O0] is contained in

a domain of type Ωb [resp., Ωa] in Diagrams (c,c). Since the concentration is not an equilibrium for

0 < τ < τ c, point O [resp., O0] is contained in a domain of type Ωa or Ωc [resp., Ωb or Ωd] in Diagram

(c), which proves (iv). Q.E.D.

Under the decomposition of domain M given by Lemma 3, and the migration pattern in each type of

domains given by Lemma 2, we can prove statement (iv) of the proposition:

Lemma 4 No cycle is possible.

Proof : First, by Lemma 2, it is evident that a cycle is impossible in areas A0EB and AEB0 which consists

entirely of type Ωa and type Ωb domains, respectively. It is also obvious from the same lemma that there

is no cycle within a type Ωc or type Ωd area.

44In Diagrams (a,b,c) points C and D describe the evolution of the same intersections as τ changes. For instance, pointD passed O as we move from Diagram (c) to (b) as τ gets smaller. In that sense, points C and D in Diagram (c) arenot the exact extension of C and D from Diagram (a). But, here, we tried to maintain the consistency of each curve-segment in terms of the adjustment process surrounding these curves. Note that the division of domain types by curvesCE,DE,C0E,D0E are consistent throughout the diagrams.

45The possibility of having more curves like CE and DE between the depicted CE and DE cannot be rejected, due tothe non-linearity of Ω(·) in na and sa.

27

By Lemma 3(ii) and (iii), area AEB consists of a type Ωc domain and isolated domains of type Ωa

and/or type Ωb. Suppose there exists a cycle passing a point in a type Ωa area within AEB. Then the

dynamic trajectory starting the point must exit the type Ωa area and enter the type Ωc area before it

comes back to the same point again. Note that at the boundary of each type Ωa area surrounded by a

type Ωc area, the direction of the adjustment is between (1, sas) and (1, sas) (refer to footnote 43). But,

recall that at each point in a type Ωc area the adjustment is directed into the domain below the 45 line

through the point. It follows by the continuity of the adjustment process that in order for the trajectory

to enter again the same type Ωa area it left, there must be a period in which the adjustment in the type

Ωc area is directed into the domain above the 45 line through the point: a contradiction. Hence, a cycle

is not possible in area AEB. The impossibility of a cycle in the case of A0EB0 can be similarly proved.

Q.E.D.

Using the above lemmas, we can describe the global dynamics of (na, sa) as presented in Proposition

3. In the following, we prove all of its statements.

Proof of statement (i):

By Lemmas 2 and 3, any segmented distribution cannot be an equilibrium in this range of τ . And by

Proposition 1 and Lemma 4, we know that there is neither equilibria (except E) nor cycles in area CED

and C 0ED0, and by the continuity of the migration process (13), any dynamic trajectory starting in these

areas eventually either reaches E, or enters area CED0 or C0ED. We also know that the adjustment is

directed toward the northeast [resp., southwest] in area CED0 [resp., C0ED]. Then, by the continuity of

the adjustment process, it follows that a dynamic trajectory starting from any point in area CED0 [resp.,

C 0ED] eventually reaches the concentrated equilibrium at O0 [resp., O]. Hence, besides the unstable

symmetric equilibrium, we have two (stable) concentrated equilibria, and no other equilibrium. Q.E.D.

Proof of statement (ii):

In this case, we have two possibilities, one in which the segment AO [resp., A0O0] of the lower [resp.,

upper] Lorenz curve is contained entirely in a domain of type Ωb [resp., Ωa] (Diagram (c)), the other in

which a part of the segment is contained in a domain of type Ωc [resp., Ωd] (Diagram (c)).46

In the former case, it can be easily shown that the global dynamics is qualitatively the same as that

for τ ∈ (bτ , 1) derived above. In the latter case, however, we have a different adjustment process. Anexample is shown in Diagram (c). Here, curve FG [resp., F 0G0] has the same property as curve DE

[resp., D0E], where the size of the segment DF [resp., D0F 0] is of strictly positive measure (refer to the

proof of Lemma 3 for the construction of curves DE and D0E). Note that in the interior of the segment

46Here, it is possible to have a segment along AO which is contained in domain of type Ωa. But, in that case, we canalways Þnd another segment along AO which is contained in a type Ωc area. To see this, note that along AO, Y < 1, whichmeans Ω(s) is increasing in s. It in turn implies that type Ωa and Ωb areas cannot be adjacent, i.e., we need to have a typeΩc area between them if Ωa should appear between A and O. Similarly, if there is a segment contained in a domain of typeΩb along A

0O0, then we also have another segment contained in the Ωd domain along A0O0.

28

DF [resp., D0F 0], there is at least one contiguous segment of strictly positive measure which is contained

in a domain of type Ωc [resp., Ωd], since otherwise the entire segment must be contained in a domain of

type Ωb [resp., Ωa] which is the case of Diagram (c).

Since the adjustment is toward the southwest [resp., northeast] on the segments OF and AD [resp.,

O0F 0 and A0D0] of the lower [resp., upper] Lorenz curve, then by the continuity of u and that of the

adjustment process, it must be true that in the type Ωc [resp., type Ωd] part of DF [resp., D0F 0], the

adjustment must change directions at least twice. It follows that when the adjustment is restricted to be

on the Lorenz curve, we have at least one stable and one unstable segmented distributions in this part

of DF [resp., D0F 0]. But, by the adjustment direction in a domain of type Ωc [resp., type Ωd] stated in

Lemma 2, we can deduce that the adjustment is directed toward this type Ωc [resp., type Ωd] part of DF

[resp., D0F 0] in its neighborhood. Hence, the segmented distributions which are stable on DF and D0F 0

are indeed stable equilibria.

It can be shown in a similar manner as in the case of bτ < τ < 1 above that a dynamic trajectory

starting from any point in domain M eventually reaches either the segmented equilibria on DF or D0F 0,

or the concentrated equilibria at O and O0, or the symmetric equilibrium at E. Hence, in this case, we

have the concentrated equilibria and at least a stable and an unstable segmented equilibria on DF and

on D0F 0. The number of segmented equilibria may differ depending on the parameter values. Though

we cannot formally show under what conditions the two cases (c) and (c) arise, our simulations indicate

that the coexistence of segmented equilibria and concentrated equilibria (i.e., (c) is more likely when the

heterogeneity of skill levels is smaller so that we have a smaller feasible domainM . It is due to the fact that

the standard core-periphery model has both concentrated and symmetric equilibria simultaneously for the

intermediate range of τ . When our model is close to the core-periphery model (i.e., small interpersonal

skill inequality), then we have also equilibria which bifurcated from these. But, as we depart from the

core-periphery model further, we may loose some of these equilibria. Also, it is not surprising given the

fact that the migration direction at each (na, sa) does not basically depend on the size of M as long as it

is in the interior of M . Thus, it is more likely for the curve ED0 [resp., ED] in Diagram (c) to hit O0A0

[resp., OA] on the upper [resp., lower] Lorenz curve if the two Lorenz curves locate closely (i.e., the size

of M is smaller). Q.E.D.

Proof of statement (iii):

Diagram (b) corresponds to this case. By Lemma 1, we know that for this value of τ , the concentration

cannot be an equilibrium. It follows that O [resp., O0] must be in the interior of areas different from type

Ωa [resp., Ωb]. This also means that D [resp., D0] cannot be on the upper [resp., lower] Lorenz curve as

C [resp., C0] (refer to the proof of Lemma 3 for the construction of curves CE and C0E). By similar

reasoning used for the case shown in Diagram (c) in the proof of statement (ii) above, we can show that

29

there is at least a stable segmented equilibrium on the segments OD and on O0D0 on the Lorenz curve.

It is to be noted that unlike the case of Diagram (c), we do not necessarily have an unstable segmented

equilibrium. Along the segment OA of the lower Lorenz curve, we may have the transition of domains

from Ωa to Ωc to Ωb, or from Ωc to Ωb. Note that adjustment along OA must change directions at least

once, but not necessarily more. Thus, we can only know the existence of one stable segmented equilibria

on the Lorenz curve. Similar reasoning applies on the segment O0A0 on the upper Lorenz curve. Q.E.D.

30

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32

high school

at most junior high school

Figure 1. Lorenz curves of inter-city inequality in education level

city order

accum

ula

ted s

hare

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

community / technical college

college

population

0 1

1

Sa

na

E

O

O’

Figure 2. Interpersonal distribution of skills and

feasible interregional distribution of population and skill

feasible domain M

B'

(a) Low transport costs ( t < t )

(b) High transport costs ( t < tc )

X=1A

A'

B

E

0 1

sa

na0 1

1

O

O'

^

X=1

O

A

B'

E

A'

B

P=1

P=1

O'

possible range of stable equilibria

1

na

sa

possible range of stable equilibria

Figure 3. Global dynamics of interregional population and skill distribution

(b) Relation between interpersonal inequality and

skill agglomeration.

(a) Relation between interpersonal inequality and

population agglomeration.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

τ = 0.8525

τ = 0.8000

τ = 0.8500

τ = 0.8513

0.4

0.5

0.6

0.7

0.8

0.9

1

25 30 35 40 45 50

τ = 0.8525

τ = 0.8000

τ = 0.8500τ = 0.8513

τ = 0.8500

500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

40

τ = 0.8525

τ = 0.8000τ = 0.8513

τ = 0.8500

τ = 0.8500

25 30 35 40 45

1

τ = 0.8000

sana

s

Figure 5. Agglomeration and interpersonal inequality when the aggregate skill in the economy is constant

( s = 25.0, σ = 5.0, c = 0.05)^

(b) Relation between interpersonal inequality and

skill agglomeration.

(a) Relation between interpersonal inequality and

population agglomeration.

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

τ = 0.8525

τ = 0.8000

τ = 0.8500

τ = 0.8513

Figure 4. Agglomeration and interpersonal inequality when the interpersonal inequality is associated with

the growth of aggregate skill in the economy ( s = 1.0, σ = 5.0, c = 0.05) _

s

τ = 0.8500

sana

s_

s_

Range of possible directionof in-migration

Range of possible direction of out-migration

(a) Domain of type Wc

0

(1,z)

(1,zo)

(1,s)

(sa,na)

na

sa

-

(1,s)-

(1,s)-

(1,z)

(1,s)-

(1,zo)

na0

(sa,na)

Range of possible directionof in-migration

Range of possible direction of out-migration

sa

(b) Domain of type Wd

Figure 6. Adjustment direction of (na,sa)

Ω bΩb

Ωb

Ω Ωa b, ΩcΩ Ωa b, Ωd

ΩaΩa

Ωa

Ωb

Ωb

Ωb

Ω Ωa b, Ω c

Ω Ωa b, Ωd

Ωa

Ωa

Ωa

Ωb

Ωb

Ωb

Ωa

Ωa

Ωa

Ω Ωa b, Ωc

Ωa Ωc

Ω Ωa b, Ωd

Ωb Ωd

Ωb

Ωb

Ωb

Ωb

Ω Ωa b, Ωc

Ω Ωa b, Ωd

Ωa

Ωa

Ωa

Ωa

(a) Low transport cost ( t < t < 1 ) (b) High transport cost ( t < tc )

(c) Intermediate transport cost ( tc < t < t ) with large skill heterogeneity

(c’) Intermediate transport cost ( tc < t < t ) with small skill heterogeneity

, or

, or

or

or

A

A'

B

B'

C

C'

DD'

O

O'

0

1

1

0

A

A'

B

B'

C

C'

D

D'

O

O'

1

1

sa

na

^

P=1

X=1

or

or

A

A'

B

B'

C C'

D

D'

O

O'

0

1

1na

sa

X=1

na

F

F'

G

G'

sa

, or

, or

P=1

P=1

X=1A

A'

B

B'

C

C'D

D'

O

O'

1

1

P=1

X=1

sa

0

E

E

E

E, or, or

na

^ ^

Figure 7. Decomposition of the feasible domain M, and global dynamics of interregional population/skill distribution